10.1.3What if the series alternates?

The Alternating Series Test

10-24.

Consider the series $S _ { n } = \displaystyle\sum _ { k = 0 } ^ { n } \frac { ( - 1 ) ^ { k } } { k + 1 }$ .

Write out $S_6$ in expanded form.
On graph paper, plot $S_n$ as a function of $n$ .
Consider the infinite series $S _ { n } = \displaystyle\sum _ { k = 0 } ^ { n } \frac { ( - 1 ) ^ { k } } { k + 1 }$ . Can you use the Divergence Test to determine if it diverges, or not? Explain your answer.

10-25.

THE ALTERNATING SERIES GAME

Let’s play a game! Split into two teams: Team Divergent and Team Convergent. Team Divergent will win one point if the outcome reveals a divergent series and Team Convergent will win one point if the outcome reveals a convergent series. You will play five rounds. The team that wins the most rounds will be champion!

Game Set Up: Sketch a long number line from $–3$ to $3$ . This will be the game board. Place a small game piece on $0$ .

How to Play: Each round will feature a different infinite series, shown below. Teams will take turns having control of the game piece. Choose which team will go first. The team in control will determine the first term of the series, and move the game piece to that location on the number line. The next team will determine the second term of the series, then move the game piece to the new location by adding (or subtracting) that value from its current location. Play continues back and forth until both teams agree that the series is convergent or divergent, and the winning team is rewarded a point.

Round 1: $\displaystyle\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n } \frac { 2 n + 1 } { n ^ { 2 } }$

Round 2: $\displaystyle\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n } \frac { n } { n + 2 }$

Round 3: $\displaystyle\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n + 1 } \frac { 2 ^ { n } } { n ! }$

Round 4: $\displaystyle\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n + 1 } \frac { n ^ { 2 } } { n ^ { 3 } + 1 }$

Round 5: $\displaystyle\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n } \frac { 1 } { n }$

10-26.

In Lesson 10.1.2, it was revealed that the inverse of the Divergence Test does not hold true. In other words, knowing that the $n^{th}$ term of an infinite series approaches zero does not guarantee that a series converges. However, the game you just played demonstrated that this inverse does apply if the series has terms that alternate between positive and negative. This is called an alternating series.

Examine the five rounds of the game. Compare the $n^{th}$ term of each series. Then copy and complete the statement below to write a conjecture about the divergence or convergence of an infinite alternating series.

The Alternating Series Test

For some alternating series $S = \displaystyle\sum _ { k = 0 } ^ { \infty } ( - 1 ) ^ { k } a _ { k }$ , if __________, then $S$ _________.

10-27.

What about the inverse of your Alternating Series Test conjecture?

Write the inverse of your Alternating Series Test conjecture. Is it always true as well? What do you notice?
Use the Alternating Series Test to determine which of the alternating series below converges. Justify your answers.
1. $S = \displaystyle \sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n - 1 } \frac { n } { n + 1 }$
1. $\quad S = \displaystyle\sum _ { i = 1 } ^ { \infty } ( - 1 ) ^ { i } \frac { 1000 } { i }$
1. $\quad S = \displaystyle\sum _ { j = 0 } ^ { \infty } \frac { ( - 2 ) ^ { j } } { j }$

10-28.

Revisit the series in problem 10-2. Identify which series are examples of alternating series. Then, for each alternating series, use this new test to determine if the series converges.

Review and Preview problems below

10-29.

Compute without a calculator Without a calculator, calculate the volume generated when the region bounded by $f(x) = x \cos(x) + 3$ and the $x$ -axis for $0 ≤ x ≤ π$ is rotated about the $y$ -axis. Then, check your answer with a calculator. Homework Help ✎

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10-30.

Evaluate each limit below. Homework Help ✎

$\lim\limits _ { x \rightarrow 0 } \frac { \operatorname { sin } ^ { - 1 } ( x ) } { x }$

$\lim\limits _ { x \rightarrow 0 } \frac { \operatorname { tan } ( a x ) } { x }$

$\lim\limits _ { x \rightarrow \infty } ( x - \sqrt { x ^ { 2 } - 1 } )$

$\lim\limits _ { x \rightarrow 0 } \frac { 1 } { x } - \operatorname { csc } ( x )$

10-31.

Determine the convergence or divergence of each of the following integrals. Homework Help ✎

$\int _ { - \infty } ^ { - 1 } \frac { 1 } { \sqrt { 2 - x } } d x$

$\int _ { - \infty } ^ { \infty } y ^ { 3 } d y$

$\int _ { 0 } ^ { 3 } \frac { 1 } { \sqrt { x } } d x$

$\int _ { 0 } ^ { \pi } \operatorname { sec } ( x ) d x$

10-32.

Calculate the arc length of the curve $x^{2/3} + y^{2/3} = 1$ . Homework Help ✎

10-33.

Use Euler’s method with a step size of $h = 0.1$ to approximate $y(0.5)$ , where $y^\prime = x^2 + y^2$ and $y(0) = 1$ . Use the second derivative to determine whether this approximation is an underestimate or an overestimate. Homework Help ✎

10-34.

Show that the curve whose parametric equations are $x(t) = t^ 3 - 3t$ and $y(t) = t ^2$ intersects itself at $(0, 3)$ . Write the equations of the two tangent lines at the point of intersection. Homework Help ✎

10-35.

For each of the following differential equations, answer the questions below. 10-35 HW eTool (Desmos). Homework Help ✎

i. When is $y(t)$ changing the fastest?
ii. What is $\lim\limits _ { t \rightarrow \infty } y ( t )$ ?
iii. Sketch a slope field.
1. $\frac { d y } { d t } = 2 y ( 1 - y )$
1. $\frac { d y } { d t } = 2 y - y ^ { 2 }$

Calculus Third Edition

The Alternating Series Test